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Examples
In Example 3 algebraic conditions specifying $D$ suggested how to write the integral as a repeated integral. Other times algebraic conditions are best interpreted graphically before deciding on limits of integration.
In Example 4 it was simpler to fix $x$ and integrate first with respect to $y$ because the bounding curves were already given to us as $$y \ = \ f_1(x)\ = \ x^2\,,\qquad y \ = \ f_2(x)\ = \ x\,.$$ Had they been given as $x \,=\, g_1(y)$ and $x\,=\, g_2(y)\,$ it might have been easier to think of $D$ as a Type II region. We would fix $y$ and first integrate with respect to $x$.
